Computing Drain Spacings

COMPUTING

DRAIN SPACINGS

C O M P U T I N G D R A I N S P A C I N G S \

Bulletin 15.

C O M P U T I N G D R A I N S P A C I N G S A generalized method with special reference to sensitivity analysis and geo-hydrological investigations

W. F. J. van BEERS Research Soil Scientist International Institute,for Land Reclamation and Improvement

INTERNATIONAL INSTITUTE FOR LAND RECLAMATION A N D IMPROVEMENT/ILRI P.O. BOX 45 WAGENINGEN T H E NETHERLANDS 1976

In memory of

Dr S. B. Hooghoudt (7 1953)

@ International Institute for Land Reclamation aiid Improvement/ILRI, Wageningen, The Netherlands 1976

This book or any part thereof must not be reproduced in any form without the written permission of ILRI

Acknowledgements

This Bulletin may be regarded as one of the results of close co-operation

between three institutions in The Netherlands which are engaged in drainage

research. Many specialists have thus contributed in one way or another to this

Bulletin, although the author bears the final responsibility for its contents.

The author's grateful acknowledgements are due in particular to:

Institute f o r Land and Water Management Research:

Dr.L.F.Ernst, Senior Research Scientist, Dept.special Research (Physics)

Dr.J.Wesseling, Head Dept.Hydrology

Agricultural University, Dept. of Land Drainage and Improvement, Wageningen:

Dr.W.H.van der Molen, Professor of Agro-Hydrology

Dr.J.W.van Hoorn, Senior Staff Scientist

International Institute fo r Land Reclamation and Improvement:

Ir.P.J.Dieleman, Land Drainage Engineer, FAO, Rome (1971)

Ir.J.Kessler, Drainage Specialist ('i 1972)

Dr.N.A.de Ridder, Geohydrologist

1r.C.L.van Someren, Drainage Specialist

5

Contents

S E C T I O N 1

S E C T I O N 2

S E C T I O N 3

S E C T I O N 4

A P P E N D I X A

A P P E N D I X B

A P P E N D I X C l

A P P E N D I X C 2

INTRODUCTION

P R I N C I P L E S O F THE HOOGHOUDT EQUATION

2.1 D i t c h e s r eaching an i m p e r v i o u s floor 2 . 2 D i t c h e s o r p i p e d ra ins l o c a t e d

above an i m p e r v i o u s l a y e r

P R I N C I P L E S O F THE ERNST EQUATION

3.1 3 . 2 T h e genera l ized o r t h e H o o g h o u d t - E r n s t e q u a t i o n 3.3 T h e m o d i f i e d H o o g h o u d t - E r n s t e q u a t i o n 3 . 4 T h e s i m p l i f i e d H o o g h o u d t - E r n s t e q u a t i o n

T h e o r i g i n a l d r a i n spac ing equat ion of E r n s t

A P P L I C A T I O N O F THE GENERALIZED EQUATION AND CORRESPONDING GRAPHS

4 .1 D r a i n a g e s i t u a t i o n s 4 . 2 S u m m a r y of g raphs and equat ions 4 . 3 P r o g r a m m e s S c i e n t i f i c Pocket C a l c u l a t o r

D E R I V A T I O N O F THE GENERALIZED EQUATION OF ERNST

LAYERED SOIL BELOW D R A I N S

CONSTRUCTION OF GRAPH 11, BASED ON HOOGHOUDT's TABLE FOR r = 0.10 m

CONSTRUCTION OF GRAPH 11, BASED ON THE GENERALIZED EQUATION O F ERNST FOR K, = K 2

L I S T O F SYMBOLS

REFERENCES

ANNEXES GRAPHS I , I a , 11, X I 1

7

I O

I O

12

i4

14 15 17 18

19

19 37 3 8

39

41

4 3

44

46

47

6

1. Introduction

This Bulletin summarizes the latest developments that have taken place in

The Netherlands on the subject of computing drain spacings using drainage equa-

tions, based on the assumption of steady-state conditions. Those based on non-

steady state conditions will be handled in a separate bulletin.

It is assumed that the reader is familiar with drainage equations in general

and with those developed in The Netherlands in particular (S.B.Hooghoudt en L.F.Ernst). These earlier contributions to the theory and practice of drainage

equations have been summarized by Van Beers (1965), who dealt specifically with

Dutch efforts in this field, and by Wesseling (1973), who also included methods

developed in other countries.

To avoid the need to consult those earlier publications, the main principles

of the Hooghoudt equation are given in Section 2 and those of the Ernst equation

in Section 3.

When using equations based on steady-state conditions,one should realize that

such conditions seldom occur in practice. Nevertheless the equations are extre- mely useful, because they make it possible:

* to design a drainage system which has the same intensity everywhere even

though quite different hydrological conditions (transmissivity values) occur in the area

* to carry out a sensitivity analysis, which gives one a good idea of the

relative importance of the various factors involved in the computations

of drain spacings.

Drainage equations and nomographs: past and present

The equations and graphs that have been available up to now are useful for the

"normal" drainage situation. By "normal", we mean that there is only one per-

vious layer below drain level and only a slight difference between the soil per- meability above drain level (K ) and that below drain level (K2).

1

7

Most equations and graphs have their shortcomings. In the following drainage

situations, for instance, there is only one possible equation that can be used:

A i m o f this bu

Because o

a highly pervious soil layer above drain level and

a poorly pervious soil layer below drain level: o n l y Eq.Hooghoudt

a heavy clay layer of varying thickness overlying a sandy

substratum: only Eq. Ernst

the soil below drain level consists of two pervious layers,

the lower layer being sand or gravel (aquifer): only Eq. Ernst

( T h i s a eommon oeeurrenee in drainage and 7:s highZy significant fo r the design.)

l e t i n

these shortcomings and the inconvenience of working with different

equations and graphs, the question was raised whether a simple equation with a

single graph could be developed to replace the existing ones. The problem was

solved by Ernst ( 1 9 7 5 ) , who combined the Hooghoudt equation and the Ernst equa-

tion for radial flow, resulting in a single expression which we shall call the

Hooghoudt-Ernst equation.

Although the fundamentals of the equation have been published elsewhere by

Ernst, it is the aim of this bulletin to focus attention on these recent de-

velopments and to illustrate the practical use of the equation and the correspond-

ing graph which has been developed for this purpose (Graph I). The graph can be

used for all the above drainage situations, although for the third one ( K 3 > > K 2 ) ,

an additional auxiliary graph will be needed.

It will be demonstrated that no graph at all is needed for most drainage si-

tuations, especially if one has available a Scientific Pocket Calculator ( S P C ) .

Although not strictly necessary, a special graph has nevertheless been prepa-

red for normal drainage situations and the use of pipe drains (Graph 11). The

reader will find that it gives a quick answer to many questions.

It may be noted that with the issue of this bulletin (No.15), a l l graphs contained in Bulletin 8 are now out of date, although Graph 1 of Bulletin 8

(Hooghoudt, pipe draj.n) still remains useful for theoretically correct computa-

tions and for the K >> K 2 situation; in all other cases Graph 2 of the present

bulletin is preferable. 1

8

The reader will also note that in this new bulletin, a revised nomenclature fo r various K- and D-UaZues (thickness of layer) has been introduced.

The modified meanings of these values are not only theoretically more correct

but also promote an easier use of the K- and D-values.

Last but not least, the importance of geo-hydrologica2 investigations, espe- cially in irrigation projects, is emphasized because a drain spacing can be con- siderably influenced by layers beyond the reach of a soil auger.

Sensitivity analysis

The primary function of a drainage equation is the computation of drain spa-

cings for drainage design. Since it summarizes in symbols all the factors that

govern the drain spacing and the inter-relationship of these factors, it also

allows a sensitivity analysis to be performed if there is a need to.

A sensitivity analysis reveals the relative influence of the various factors

involved: the permeability and thickness of the soil layers through which ground-

water flow can occur (depth of a barrier), wetted perimeter of drains, depth of

drains, etc. This analysis will indicate whether approximate data will suffice

under certain circumstances or whether there is a need for more detailed investi-

gations. The drainage specialist will find the sensitivity analysis a useful

tool in guiding the required soil and geohydrological investigations,which differ

from project to project, and in working out alternative solutions regarding the

use of pipe drains or ditches, drain depth, etc.

For a sensitivity analysis, however, it is a "conditio sine qua non" that

the available equations and graphs should be such that the required calculations

can be done easily and quickly.

In the opinion of the author, this condition has been fulfilled by the equa-

tions and graphs that will be presented in the following pages,especially if one

has an SPC at his disposal.

9

2.

2.1

Principles of the Hooghoudt equation

Ditches reaching an impervious floor

For flow of groundwater to horizontal parallel ditches reaching an impervious

floor (Fig.1) horizontal flow only, both above and below drain level may be

assumed, and the drain discharge, under steady-state conditions can be computed with a simple drainage equation:

8K2D2h 4Klh2 q=-+- or ( 1 )

L2 L2

where

q

q2 = discharge rate for the flow below drain level q1 = discharge rate for the flow above drain level D2 = thickness of the pervious soil layer below drain level (m)

= drain discharge rate per unit surface area per unit time (m3 per day/m2 or miday)

(depth to an impervious layer or depth of flow) or

per unit length (metre) of drain (m2/m) = cross-sectional area of flow at right angles to the direction of flow

K2 = hydraulic conductivity of the soil (flow region) below drain level

K 1 = hydraulic conductivity of the soil (flow region) above drain level

h

(m/ day)

(m/day); for homogeneous soils K

way between drains (m); note that the water table is defined as the locus of points at atmospheric pressure

1 = K2 = hydraulic head - the height of the water table above drain level mid-

L = drain spacing (m)

If, for the flow above the drains, one wants to avoid the use of a certain

notation (h) for two quite different factors, being a hydraulic head and an

average cross-section of flow area (4 h), it is preferable to write Eq.(l) as

8KzD2h 8KlDlh (Fig. I ) q = 2 + - - L L2

where

D, = average depth of flow region above drain level or average thickness of the soil layer through which the flow above the drains takes place.

IO

F i g . 1 . Cross-sections of flow area. Steady-state conditions: discharge ( q ) = recharge ( R ) . Parallel spaced drains reaching an impervious f l oor .

Various discharge values:

day. When the drain spacing is 40 m, the discharge per metre of drain is qL = 0.005 X 40 = 0.2 m3 per day; when this drain is 100 m long, the discharge of the drain will be 0 . 2 x 100 = 20 m3 per day or 20,000/86,400 = 0.23 litres

per sec., and in this case the discharge per ha will be 0.23 x 10,000/(40~100)

= 0.58 lit.Sec.ha.

Note that 1 lit.Sec.ha = 8.64 mm per day or 1 mm per day = 0.116 lit.Sec.ha.

q = 0.005 m/day = 0.005 m3 per m2 area drained per

In comparison with Eq.(l), Eq.(la) shows more clearly that the discharge

rate for the flow above and below the drains can be computed with the same

horizontal flow equation; the only difference is the cross-section of flow area.

Eq.(la) can be used for a drainage situation with two layers of different

permeability (K2 and KI), drain level being at the interface of these layers.

The equation can also be used for homogeneous soils (K =K ) . This is possible because Hooghoudt distinguishes primarily not soil layers but groundwater flow regions, split up into a flow above drain level and a flow below drain level. These flow regions can coincide with soil layers but need not necessarily do so.

2 1

2.2 Ditches or pipe drains located above an impervious layer

When ditches or pipe drains are located above an impervious layer, the flow

lines will be partly horizontal, and partly radial as they converge when approach-

ing the drains. This causes a restriction to flow (resistance to radial flow)

due to a decrease in the available cross-section of flow area. The smaller the

wetted perimeter of the drain, the greater the resistance to radial flow.

In certain respects, this flow restriction can be compared to the traffic on

a highway, where in a certain direction one of the lanes is blocked. In both

cases, the available cross-sectional flow area has been reduced.

For the drainage situation described above, the Hooghoudt equation expressed

in terms of Eq.(la) reads

8Kndh 8K1Dlh

q = - T + - L L2

where

d = the thickness of the so-called "equivalent layer" which takes into

account the convergence of flow below the drain (q2) (radial flow) by

reducing the pervious layer below the drain (D2) to such an extent that the horizontal resistance (%) plus the radial resistance (R,) of the

layer with a thickness D2 equals the horizontal resistance of the layer

with a thickness d. This d-value is a function of the drain radius (r),

L- and D -value (Hooghoudt's d-tables). 2

If we compare Eq.(la) with Eq.(2), it can be seen that both are horizontal

flow equations. The only difference being that the D -value of Eq.(la) (horizon-

tal flow only) has been changed into a d-value in Eq.(2) (horizontal and radial 2

flow).

Summarizing, the main principles of the approach of Hooghoudt are:

( 1 ) Primarily, he distinguishes groundwater flow regions, split up into flow above and beZow drain level, and only secondarily does he distin- guish soil layers;

(2) for the flow region above drain level, only horizontal flow need be con-

sidered (transmissivity K D ) , whereas for the flow region below drain

level, both horizontal flow (transmissivity K D ) and radial flow have

to be taken into account;

1 1

2 2

12

( 3 ) t h e r a d i a l f low i s accounted f o r reducing t h e depth D2 t o a s m a l l e r

depth d , t h e so-ca l led e q u i v a l e n t l a y e r .

The adopt ion of t h e f i r s t p r i n c i p l e r e s u l t e d i n a uniform nomenclature f o r

t h e Hooghoudt and E r n s t equat ions and t h e use of t h e same equat ions f o r homogeneous

s o i l and a s o i l w i t h t h e d r a i n s a t t h e i n t e r f a c e of two l a y e r s .

The adopt ion of t h e second p r i n c i p l e r e s u l t e d i n a change i n t h e o r i g i n a l

E r n s t e q u a t i o n , w i t h r e s p e c t t o t h e magnitude of t h e r a d i a l f low.

The t h i r d p r i n c i p l e , d e a l i n g w i t h t h e mathematical s o l u t i o n of t h e problem

of r a d i a l f low, has been changed. I n s t e a d of changing t h e D -value i n t o

a s m a l l e r d-value - t h e r a d i a l f l w h a s been taken i n t o account by changing a

L -value ( d r a i n spac ing based on h o r i z o n t a l f low o n l y , L: = 8KDh/q) i n t o a s m a l l e r

L-value ( a c t u a l d r a i n spac ing based on h o r i z o n t a l and r a d i a l f l o w ) .

2

This a l t e r n a t i v e s o l u t i o n r e s u l t s i n one b a s i c d r a i n a g e equat ion and only

one g e n e r a l graph t h a t can be used f o r a l l d r a i n a g e s i t u a t i o n s , p i p e d r a i n s

a s w e l l a s d i t c h e s , and a l l d r a i n s p a c i n g s , wi thout having t o u s e d- tab les o r

a t r i a l - a n d - e r r o r method o r s e v e r a l g raphs .

13

3. Principles of the Ernst equation

3.1 The original drain spacing equation of Ernst

The general principle underlying Ernst's basic equation (1962) is that the

flow of groundwater to parallel drains, and consequently the corresponding avail-

able total hydraulic head (h), can be divided into three components: a vertical

(v), a horizontal (h), and a radial (r) component or

where q is the flow rate and R is the resistance.

Working out various resistance terms, we can write the Ernst equation as

aD2 h = q - + DV q-+ L2 q-In- 8KD nK2 KV

where

(3)

h, q, K2, D2, L = notation Hooghoudt's equation (Section 2.1)

Dv = thickness of the layer over which vertical flow is considered; in most

cases this component is small and may be ignored (m)

K = hydraulic conductivity for vertical flow (m/day)

KD = the sum of the product of the permeability (K) and thickness (D) of the various layers for the horizontal flow component according to the

hydraulic situation:

one pervious layer below drain depth: KD = K D two pervious layers below drain depth:KD = KIDl + K2D2 + K3D3

V

+ K2D2 1 1 (Fig. 2a)

(Fig.2b) a = geometry factor for radial flow depending on the hydraulic situation:

KD = K D KD = KIDl + K2D2 + K3D3, the a-value depends on the K2/K3 and D2/D3

+ K2D2, a = I 1 1

ratios (see the auxiliary graph Ia)

u = wetted section of the drain (m); for pipe drains u = nr.

Eq.(3) shows that the radial flow is taken into account for the total flow (q),

whereas the Hooghoudt equation considers radial flow only for the layer below

drain level (q2).

It should be noted that Eq.(3) has been developed for a drainage situation

where K << K2 I used where the flow above drain level is relatively small. However, if one uses

14

( a clay layer on a sandy substratum) and can therefore only be

this equation for the case that K uncommon situation), the result is a considerable underestimate of drain spacing compared with the result obtained when the Hooghoudt equation is used.

>> K2 (a sandy layer on a clay layer, a not I

According to Ernst (1962) and Van Beers (1965), no acceptable formula has

been found for the special case that KI >> K2, and it was formerly recommended

that the Hooghoudt equation be used for this drainage situation. Since that

time, as we shall describe below, a generalized equation has been developed,which

covers all K /K ratios. 1 2

K 3 C0.l K2 K3D3 neglible

F i g . 2 a . S o i l below t h e d r a i n : o n l y o n e p e r v i o u s l a y e r ( D 2 )

K2D2) (KD = KIDl +

F i g . 2 b . S o i l below the d r a i n : two p e r v i o u s l a y e r s (D2, D3)

(KD = KIDl + K 2 D 2 + K3D3)

. . . . i m a g i n a r y boundary - - - - r e a l boundary

Fig.2.Geometry of the Ernst equation if the vertical resistance may be ignored.

3.2 The generalized or the Hooghoudt-Ernst equation

This new equation is based on a combination of the approach of Hooghoudt

(radial flow only for the flow below the drains, q2) and the equation of Ernst

for the radial flow component.

Neglecting the resistance to vertical flow and rewriting Eq.(3), we obtain

8KDh

If we consider only horizontal flow above the drains (q ) and both horizontal 1 and radial flow below the drains (q2), we may write

15

q=-+ aD 2

L * + - D LI^- 8 L2 TI

( 4 )

Introducing an equivalent drain spacing (L ),i.e. a drain spacing that would

be found if horizontal flow only is considered, we get

2 - 8KDh Lo - - 4

Substituting Eq.(5) into E q . ( 4 ) yields

8KDh 8KiDlh 8KzD2h

(5)

After a somewhat complicated re-arrangement (see Appendix A) the generalized

equation reads

where

L = drain spacing based on both horizontal and radial flow (m)

Lo = drain spacing based on horizontal flow only (m)

c = Dz In - , a radial resistance factor (m) aD 2

K i D i

= KD B = the flow above the drain as a fraction of the total horizontal

flow

GRAPH I

To avoid a complicated trial-and-error method as required by the Hooghoudt

approach, Graph I has been prepared. For different c/Lo and B-values, it gives the corresponding L/L -value which, multiplied by L O' gives the required drain

spacing (L-value). However, as will appear furtheron, this graph is usually

only needed for the following specific situations:

16

K, >> K2 or B > 0.1;

there are two pervious layers below drain level, K3D3 > K2D2, and

no Scientific Pocket Calculator (SPC) is available;

one wants to compare the generalized equation with other drainage

equations or one wants to prepare a specific graph ( s e e Section 4.1 and Appendix C 1 ) .

NOTE: Comparing the Hooghoudt equat ion u i t h t h e E r n s t equat ion and using

u = 4r ins t ead o f u = m, a grea te r s i m i t a r i t y is obta ined .

3.3 The modified Hooghoudt-Ernst equation

In most actual situations the B-factor (K D /KD ratio) will be small and there-

the last term in Eq.(7) has little influence on the computed drain spacing. 1 1

fore

Neglecting B, and rewriting Eq. (7) (multiplying by L3/L and substituting

8KDh/q for L2), we obtain the equation for the B = O line of Graph I:

DP 8L L 2 + - Dz In - - 8KDh/q = O TI

If we compare Eq.(8) with the original Ernst equation (3a), it can be seen KD KP KP that the factor - of Eq. (3a) has changed into , which equals DP. However,

for a drainage situation with two pervious layers below drain level,

becomes

KzD2 - KZ K2D2 + K3D3

KP and the equation f o r this situation reads

8L KZDP + K3D3 aDs In - - 8 KD h/q = O L2 + -

TI KP

A s regards the use of Eq.(9) it can be said that, in practice, Eq.(9) is very

useful if an SPC is available; if not, Graph I has to be used.

Eq.(8), on the other hand, will seldom be used. The simple reason for this is that for most drainage situations with a barrier (only K D ) , a simplified

equation can be used. 2 2

17

3.4 The simplified Hooghoudt-Ernst equation

After Graph I had been prepared on linear paper, it was found that for c/L -

values < 0.3 and B-values < 0.1, the following relations hold

LILo = 1 - CfL or L = L - c (see Graph I) (10)

The question then arose whether these conditions were normal or whether they

were rather exceptional.In practice,the simplified equation proved to be almost al-

ways applicable.In addition,it was found from the calculations needed for the prepa-

ration of Graph II(r = 0.10 m, all K- and D-values) that, except for some un-

common situations (K = 0.25 m/day, D > 5 m), the equation L = L - c is a

reliable one, where C = D2 In D . Note: Many years ago W.T.Moody, an engineer with the U.S.5ureau of Reclamation

D proposed a similar correction iD l n G) t o be subtracted from the calculated

spacing (Maasland 1956, D m 1 3 6 0 ) . The only di f ference between the correction

proposed by Moody and that i n this b u l l e t i n is that now the conditions under

which the correction may be applied are precisely defined.

18

4. Application of the generalized equation and corresponding graphs There are many different drainage situations, five of which will be handled

in this section.

SITUATION 1 :

K I = K2

SITUATION 2 :

K I 2 Ka

SITUATION 3 :

K I > > K2

SITUATION 4 :

1 2 K << K

SITUATION 5: <

2 K3 > K

Homogeneous soil; D < 4L; pipe drainage

Slight differences between soil permeability above and below

drain level; differences in depth to barrier ( D 5 iL); pipe

and ditch drainage

A highly pervious layer above drain level and a poorly

pervious layer below drain level

A heavy clay layer of varying thickness overlying a sandy sub-

stratum; the vertical resistance has to be taken in account

Soil below drain level consists of two pervious layers

( K 2 D 2 , K D ) ; the occurrence of an aquifer (K3 >> K 2 ) at

various depths below drain level. 3 3

4.1 Drainage situations

SITUATION 1: Homogeneous soil; O < &L; pipe drains

K = K The use of t he simplified equation and Graph 11

This is the most simple drainage situation; the required preparatory calcu-

lations are limited and a graph is usually not needed. For comparison with

other drain spacing equations, we shall use the example given by Wesseling (1973) .

Note that - for reasons of convenience - in this and the other examples the

units in which the various values are expressed have been omitted with the

exception of the L-value. However, for values of h, E , and 5, read metres; for q and K , read m/day and for KD, read m2/day. - -

19

PREPARATORY C A L C U L A T I O N S DATA G I V E N COMPIJTATION D R A I N S P A C I N G ( L )

L2 = KD 8h/q

h = 0.600 h/q = 300 Lo = 100.9 L = L - c = 8 7 m

q = 0.002 8h/q = 2400 c = 13.8 ~~~~

K =0.8 n =0.30 K I D l = 0.24 c < 0.3 L Eq.Hooghoudt: L = 87 m 1 1

L = 8 4 m

Eq.Kirkham: L = 85 m Eq .Dagan : L = 8 8 m

K =0.8 D =5.0 K2D2 = 4.0 B < 0.10 E q .Ernst : 2 2

r = 0.10 KD = 4.24 (Eq.10 may be

(Wesseling 1973) u = nr B = 0.06 used)

NOTE: writing h = 0.600 instead of 0.6, is not meant to suggest accuracy, but is only f o r convenience in determining the h/q value.

If no SPC is available, the c-value (D2 In -1 can be obtained from Graph 111. D2

The simplified formula is very convenient if we want to know the influence

that different U-values will have on the drain spacing. For example

r = 0.05 m, then c = 17.3 and L = 84 m u = 1.50 m, then c = 6.0 and L = 95 m

The influence of different K or D-values is also easy to find. However, if the U-value is fixed, it is better to use Graph I1 or a similar graph, which

gives a very quick answer to many questions.

GRAPH I1

This graph is extremely useful for the following purposes and where the fol-

lowing conditions prevail:

Purposes

A great number of drain spacing computations have to be made, for instance,

for averaging the L-values in a project area instead of performing one

calculation of the drain spacing L with the average K- or KD-value;

20

- One wants to find out quickly the influence of a possible error in the K-

value or the influence of the depth of a barrier (D-value);

One wants to demonstrate to non-drainage specialists the need for borings

deeper than 2 . 1 0 m below ground level because a boring to a depth of 2 . 1 0 m results in a D -value of about I m, being 2 . 1 0 m minus

drain depth. 2

Conditions

Homogeneous soil below the drains (only K D ) ;

The wetted perimeter of the drains has a fixed value, say pipe drains with

2 2

r = 0.10 m or ditches that have a certain U-value;

An error of 3 to 5% in the computed drain spacing is allowable.

Considering these purposes and conditions, it will be clear that the avail-

ability of Graph I1 or similar graph is highly desirable, except when:

exact theoretical computations are required

KI >> K2

there are two permeable layers below drain level instead of one (K D K D ) . 2 2 ' 3 3

For these three conditions Graph I1 cannot be used and one must resort to

Graph I.

Other q-, h- or KI/K2 values than those given on the graph

Graph I1 has been prepared for the following conditions:

h = 0.6 m, q = 0.006 m/day or h/q = 100, K l = K 2 m/day, and r = 0.10 m

For the specific purposes and conditions for which this graph has been pre-

pared (approximate L-values suffice) an adjustment is only required for different

h/q values, through a change in the K -values to be used. For instance, in the

example given for Situation 1 we have a h/q value of 300. Therefore 2

and we read for D = 5 m, L = 87 m.

21

However, if many computations have to be done, it is preferable to prepare

a graph or graphs for the prevailing specific situation. For instance, for the

conditions prevailing in The Netherlands, three graphs for pipe drains would be

desirable: q = 7 "/day and h = 0.3, 0.5, and 0.7 m.

For irrigation projects a q-value of 2 "/day is usually applied.

If one wants to know the magnitude of an introduced error (h # 0.6, K I # K2), the extra correction factor (f) for K; can be approximated with the formula

where

D l = 0.30 and D i = 0.5 h' .

For example:

h' = 0.90 D ; = 0.45

5 + (0 .5 x 0.45) = o.986 or 5 + 0.30 K /K2 = 0.5, D = 5 f = 1 2

about 1 % difference in the L-value

When the assumption of a homogeneous aquifer contains a rather large error (e.g.

K = 2 K ) , and moreover a larger hydraulic head being available (k'=0.9 m), we get

f=l.lI or 5% difference in the L-value. 1 2

Preparation

The preparation of a speficic graph is very simple, and takes only a few

hours. There are two methods of preparation.

Appendix C 1 gives an example of how it is done if the d-tables of Hooghoudt

are available. This is the easiest way.

Appendix C 2 shows a preparation based on the generalized equation in combina-

tion with Graph I. This method gives the same result, but requires more calcu- lations.

Finally note that Graph I1 demonstrates clearly that if, in a drainage project,

augerholes of only 2 m depth are made (D

required drain spacing can result.

2 1 m), considerable errors in the 2

22

For example:

Given: h / q = 300 ( i r r i g a t i o n p r o j e c t ) , K 2 = 0.8 , t h e n K; = 2 . 4 ;

d r a i n depth = 1.5 m

D e p t h t o b a r r i e r Flow d e p t h S p a c i n g h / q = 100, K2 = 0.5

(D2) ( L )

2.50 m I m 50 m L = 2 2 m

3.50 m 2 m 63 m 28 m

6.50 m 5 m 8 7 m

11.50 m I O m 105 m

34 m

37 m

m m 130 m 37 m

This example may show t h a t :

Graph I1 i s very s u i t e d t o c a r r y out a s e n s i t i v i t y a n a l y s i s on t h e

i n f l u e n c e of t h e depth of a b a r r i e r , e t c .

2 has been cons idered . However, t h e r e can a l s o be a c o n s i d e r a b l e change

i n t h e K -value. 2

- The need f o r d r i l l i n g deeper than 2 m. Note t h a t h e r e o n l y t h e v a l u e D

The r e l a t i v e i n f l u e n c e of t h e D -value changes w i t h t h e spac ing o b t a i n e d . 2

SITUATION 2: Slight differences between soil permeability above and below K I K 2 drain level (KI 3 K2);

Differences in depth to a barrier iD 5 1 / 4 L ) ;

pipe and ditch drainage

F o r t h e s i t u a t i o n D < {L and d r a i n a g e by d i t c h e s , t h e computation of t h e

d r a i n spac ing , as wel l a s t h e computation s h e e t i s t h e same a s have been g iven

i n S i t u a t i o n I . Only i f K I > K2 and D2 i s smal l should s p e c i a l a t t e n t i o n be pa id

t o t h e q u e s t i o n whether B < 0.10.

For t h e s i t u a t i o n D > a L , t h e fo l lowing equat ion (Erns t 1 9 6 2 ) can be used:

23

The use of this equation will be demonstrated below and will be followed by a

sensitivity analysis for the u and D2 factor.

G I V E N COMPUTATION L h = 0.800

q = 0.002 Graph 111, for u = 1.50 + L = 205 m K I = 0.40

K2 = 0.80

h/q = 400 L In - = 71 x 0.8 X 400 = 1005

If a SPC is available, the L-value can also

be obtained by a simple trial- and error-method

error method

u = 1.50

Equation (11) does not take the horizontal resistance into account because it

is negligible compared with the radial resistance; nor is the flow above the

drain considered. Only if the computed L-value is small, say about 40 m or less, is a small error introduced.

If Graph I only is available, or one wants to check the computed L-value by using the generalized equation, the following procedure can be followed:

Estimate the drain spacing and assume a value for D2 between i L and $L

(beyond this limit, the computed spacing would be too small); compare the two

L-values (control method) o r check whether the assumed D -value > 4L and < 0.5L

(computation method). 2

For the above example we get:

Given: D 2 = 80

Assume K2 = 0.8 KD = 64 Lo = 452 c/Lo = 0.70

Assume u = 1.5 8 h/q=3200 c = 318 Graph I: L/Lo = 0.45 + L = 204 m

Using a SPC and Eq.(8) -f L = 202 m

Note: It may be useful - by way of exercise - t o t r y other D compare the resulting L-ualues.

ualues and to 2

24

Sensitivity analysis for U-value and depth to a barrier (D-value)

D, = m D 2 = 5 m L

KD = 0.16 + 4.0 = 4.16 8 h/q = 3200 Lo=115 m u = l m + L = 1 9 2 m u = l m + c = 8 m + L = 107m

u = 1.5 m + L = 205 m u = 1 . 5 m c = 6 m + L = 109m u = 2 m + L = 2 1 5 m u = 2 m c = 4 . 5 m + L = 110m

u = I O m + L = 300 m

u = 0.30 m + L = 162 m u = 0.30 m + L = 101 m; u = 0.20 + L = 101 m

u = 0.40 -+ L = 103 m

u-va Zue

This sensitivity analysis shows that the influence of difference in the

U-value increases as L increases. However, differences of 50% or more are gene- rally of little importance. Therefore the U-values of pipe drains can be appro-

ximated by taking r = 0.10 m (u = 0.30) and the U-values of ditches approxima-

ted by taking the width of the ditch and two times the water depth (usually

2 x 0.30 m).

The slope of the ditch need not be taken into account because of the reasons

mentioned. Moreover, neither the water level in the ditch nor the drain width are

constant factors.

D-vazue

The large error made by assuming D = 5 m instead of D = m,as in the above 2 2 example (L = 109 m instead of 205 m), is easily made if, in an irrigation project

(q = 0.002, h/q and L-value very large), no hydro-geological investigations are

carried out.

SITUATION 3: A highZy pervious Zayer above drain Zevel and a poorly pervious Kl>>K2 Zayer beZow drain ZeveZ

The use o f Graph I

This particular situation is frequently found. It may be of interest to use the data given below to compute drain spacings with other drainage equations and

then to compare the results with those obtained with the equation of Hooghoudt

or the generalized Hooghoudt-Ernst equation.

25

DATA G I V E N PREPARATORY C A L C U L A T I O N S COMPUTATION

h = 1 .O00 h/q = 200 Lo = 53.7 m

q = 0.005 8h/q = 3200 c = 12.6 m

K = 1.6 D =0.50 KIDl = 0.8 c/L =0.235 L/Lo = 0.88 (Graph I)

L = 0.88 x 53.7=47.3 m K = 0.2 D =5.0 K2D2 = 1.0 B = 0.44 I 1

2 2

r = 0.10 KD = 1.8

u=O .40 B = 0.44 *

Note: The computation shee t used here i s t h e same as t h a t used f o r S i t u a t i o n I, except t h a t no c/L -0alue i s needed i n S i t u a t i o n 1 .

O

*For t h e o r e t i c a l comparisons w i t h t h e r e s u l t s of t h e equat ion of Iiooghoudt, it i s pre f e rab le t o use u = 4 r ins t ead o f u = T r

Comparison o f the E r n s t equat ions with t h e equat ion o f Hooghoudt

Original equation of Ernst (Eq.3a) L = 32 m

Modified equation of Ernst (Eq.8) L = 39.9 m

Generalized equation of Ernst (Eq.7) L = 47.2 m

Equation Hooghoudt (Eq.2) L = 47.2 m

Graph 1I:K; = 0.2 X 2 X 1.7 = 0.68, D2 = 5 m + L + 41 m

T.t should be born in mind that the original equation of Ernst never has been

recommended for the considered situation with a major part of the flow through

the upper part of the soil above drain level (K <<K ) . Therefore it is not sur-

prising that the unjustified use of this formula will result in a pronounced

underestimating of the drain spacing; the modified equation is somewhat better,

while the generalized equation gives the same results as the equation of

Hooghoudt. In addition,it is demonstrated that the use of Graph I1 for the KI>>K2

situation also results in an underestimate of the drain spacing.

2 1

26

SITUATION 4:

1 2

A heavy clay layer of varying thickness overZying a sandy K <<K substratwn; the v e r t i c a l resistance has t o be taken i n t o account

This is another drainage situation that occurs frequently. Because the thick-

ness of the clay layer can vary, three different drainage situations can result

(see Fig.3). In this example, it is assumed that the maximum drain depth is

-1.40 m, in view of outlet conditions, and that the land is used for arable

farming (h = drain depth - 0.50 m = 0.90 m). U V

V The computation of the vertical component (hv = q , see Section 3.1) is

somewhat complicated because the D -values varies with the location of the drain with respect to the more permeable layer. However, Fig.3 and the corresponding

calculations may illustrate sufficiently clearly how to handle the specific drai-

nage situations. It may be noted that as far as the author is aware the solution

given by Ernst for this drainage situation is the only existing one.

V

q=O.OlO m /dav

D v = h + y Dv = h D = h - D '

h" = t Dv h = q h h" = k"; Dv v % h ' = h - h h ' = h - h h ' = h - h

KD = K D t K3D3 2 2

c=:1nu aD2

KD = K2D2 KD = K i D i + K2D2

D2 c = D l n - 2 u c = D l n - D2

2 u

Ex.4a Ex.4b Ex.4c

D r a i n l e v e l above D r a i n l e v e l c o i n c i d e s D r a i n l e v e l below t h e boundary w i t h t h e boundary t h e boundary o f t h e

two s o i l l a y e r s

F i g . 3 . Geometry of t he Ernst equation i f v e r t i c a l resis tance has t o be taken i n t o account (K <( KZI. 1

27

Procedure

9 Determine D according to the specific situation;

Calculate h the loss of hydraulic head due to the vertical resistance,

V

D V '

by using h = - ;

Calculate h' from h' = h - h where h' is the remaining available hydraulic V

head for the horizontal and radial flow

Calculate the h'/q and KD-value. Note that the horizontal flow in the upper

layer with low permeability may be ignored;

Compute L; f o r Example 1 , Graph Ia is required in addition to Graph I or an

SPC and Eq.(9); for Examples 2 and 3, use L = L - c.

The following examples are intended to illustrate the procedure and the layout

of a computation sheet. (The data used have been taken from Fig.3).

Example 4a

Dv = h+y = 0.90 + 0.30 = 1.20 hv = q/K1 x Dv = 0.2 X 1.2 = 0.24

h / q = 66

8 h /q 528 h'=h-h =0.90-0.24=0.66 V

Kv = 0.05 D =0.80 K2D2 = 0.04 L0=50.6 m c / L =1.45 LILo = 0.25

K = 2.0 D =2.40 K D - 4.80 -+ c =?3.3 m B = 0 L = 12.6 2

3 3 3 3 -

K IK =20 D3/D2=3 KD = 484 3 2 U a = 4.0 ( s ee Ex.5)

AZternativas

Pipe drains (u=0.30) + L = 5 m

- Ditch bottom i n the more permeable layer (ditch depth at 2.20) +

u = 0.90 and h = 1.70 + Lo = 80 m; c = 2m -f L = 78 m Pipe drains at -2.20 m + Lo = 80 m; c = 5 m + L = 75 m

Note: The Zast t#o a l t e r n a t i v e s mean t h a t t h e drainage water w i Z l haue t o be

discharged by pumping.

28

Example 46

Dv = h = 0.90 h /q = 72 hv = 0.2 X 0.90 = 0.18 8 h/q = 576 h ' = 0.90 - 0.18 = 0.72

K2 = 2.0 D2 = 3.20 KD = 6.40 Lo = 60.7 m L = 58.3 m

c = 2.4 m

Example 4c

Dv = h - D i = 0.90 - 0.40 = 0.50

h = 0.2 x 0.50 = 0.10

K' = 2.0 D' = 0.40* K'D'=0.80 Lo = 67.9 m L = 65.5 m

h ' = 0.80 8 h/q = 640

V

1 1 1 1

K = 2.0 D2 = 3.20 K D =6.40 c = 2.40 m 2 2 2

KD = 7.20

* The available cross-section f o r horizontaZ f l o w = thickness of t he more pemeabZe Zayer above drain depth.

Remarks

I f we compare t h e computed d r a i n spac ing f o r Example 4b (L=58 m) wi th t h a t

of Example 4c (L=65 m ) , we can conclude t h a t f o r a g iven d r a i n depth t h e e x a c t

t h i c k n e s s of t h e heavy c l a y l a y e r i s of minor importance a s long as t h e bottom

of t h e d r a i n i s l o c a t e d i n t h e more permeable l a y e r . 1111

I f , however, t h e c l a y l a y e r cont inues below d r a i n d e p t h , a s i n Example 4a

(L=13 m), d r a i n spac ings would have t o be v e r y narrow indeed and t h e a r e a w i l l

s c a r c e l y be d r a i n a b l e ( f o r p i p e d r a i n s , L=5 m , f o r a d i t c h , L=13 m ) .

The only way o u t h e r e i s t o use deep d i t c h e s (L = 78 m) o r p i p e d r a i n s

(L = 75 m) t h a t reach i n t o t h e permeable l a y e r , and t o d i s c h a r g e t h e d r a i n a g e

water by pumping.

29

SITUATION 5: SoiZ below drain depth cons i s t s o f two pervious layers

(K2D2, KgDgj.

Graph I a . The occurrence o f an aquifer ( K 3 >> K2) a t v a r i o u s depths below d r a i n level

Hydrologically speaking, this drainage situation is very complicated. Up to

now the problem could only be solved by using an additional graph (Ia) 1962) or by the construction of various graphs for various drainage situations

(ToksÖz and Kirkham, 1971 ) .

(Ernst

The graph of Ernst that can be used for all situations (various K3/K2 and

D /D ratios) gives the results he obtained by applying the r e h x a t i o n method. A somewhat modified form of this graph has been published by Van Beers (1965). 3 2

The reZiabiZity and importance o f Graph I a

Considering the method by which this graph has been constructed, the question

arises as to how reliable it is. The correctness of an equation can easily be

checked, but not the product of the relaxation method.

Fortunately, the results obtained with this graph could be compared with the

results obtained with 36 special graphs, each one constructed for a specific

drainage situation (ToksÖz and Kirkham, 1971). It appeared that both methods gave

the same results (Appendix B). Thus the conclusion can be drawn that the gene- ralized graph of Ernst is both a reliable and an important contribution to the

theory and practice of drainage investigations. It is particularly useful in drai-

nage situations where there is an aquifer (highly pervious layer) at some depth

( I to 10 m or more) below drain level, a situation often found in irrigation

projects.

When there are two pervious soil layers below drain level, the two most common

2' drainage situations will be: K3<< K2, and K3>> K

Situat ion Kg << K 2

The availability of Graph Ia enables us to investigate whether we are correct

in assuming that, if Kg < 0.1 K 2 , we can regard the second layer below drain depth

(K D ) as being impervious. If we consider the L-values for this situation, as 3 3

30

given in Appendix B, we can conclude that although the layer K3D3 for K 3 < 0 . 1 K2

has some influence on the computed drain spacing, it is generally so small that

it can be neglected. However, if one is not sure whether the second layer below

drain level can be regarded as impervious, the means (equation and graph) are

now available to check it.

3 >> K 2 Situation K

This situation is of more importance than the previous one because it occurs

more frequently than is generally realized and has much more influence on the

required drain spacing.The examples will therefore be confined to this situation.

Examples ( see F i g . 4 )

Given: The soil or an irrigation area consists of a loess deposit (K = 0.50

m/day) of varying thickness. In certain parts of the area an aquifer occurs

(sand and gravel, K = 10 mlday, thickness 5 m).

In the first set of examples (A) the loess deposit is underlain by an imper-

vious layer at a certain depth, varying from 3 to 40 m. In the second set

of examples (B), instead of an impervious layer, an aquifer is found at a

depth of 3 m and 8 m, whereas Examples C give alternative solutions in rela-

tion to drain depth and the use of pipe drains instead of ditches.

It is intended to drain the area by means of ditches (drain level = 1.80 m,

wetted perimeter (u = 2 m). The maximum allowable height of the water table

is 1 m below surface (h = 0.80 m). The design discharge is 0.002 m/day

(h/q = 4 0 0 ) .

A . Influence of the loca t ion of an impervious l aye r . Homogeneous s o i l

For this simple drainage situation, only the result of the drain spacing

computations will be given.

Drain spacing Example Depth b a r r i e r

Ditch (u=2 m ) Pipe drain (u=0.30 m )

A l 3 m

A 2 8 m

A 3 40 m

50 m

96 m

146 m

49 m

84 m

107 m

31

These results show that the depth of a barrier has a great influence on the

drain spacing and that the influence of the wetted perimeter of the drains (u)

can vary from very small to considerable, depending on the depth of

DRAIN SPACINGS A N D D R A I N A G E SURVEY N E E D S IN IRRIGATION PROJECTS

Influence location of B an aquifer A an impervious layer I

1 2 3

-. - - - - - - - - -

q-O 002 m/day

loess K=050m/day

impervious layer (fine textured a I I u vial depos its )

ditch (uZ2m) L z ' 5 0 m b6m \ 4 6 m pipedrain L ~ 4 9 m 8 4 m 107m

( u I 0.30)

1

, I

I I

305 m

la

O

I I I , 1 I I

180m

2

, I I I I 6 2 5 m

O

I i I I I I I 615 m

the barrier.

, 200 m

KI= hydr. cond. above drain level 1 lntluence KD aquifer 1 I

K ~ Z ., .. below ,. ,, (first layer) 1000 m2/day 1000 m*/day K j i . . .. .. ,, .. (second layer) ' L ~ 6 2 5 m u : wetted perimeter 1 much influence l i t t ie influence I

Lz245m

Fig.4. Drain spacings and drainage survey needs in irrigation projects.

B. Influence o f t he location of an aqui fe r

The various situations that will be handled here are:

Example Depth aqui fe r Drain level Ditches Pipe drains Spacing

B I - 3 m - 1.80 m + 305 m

B la - 3 m - 1.80 m + 180 m

B 2 - 3 m - 3 m + 625 m

B Za - 3 m - 3 m + 615 m

B 3 - 8 m - 1.80 m + 200 m

32

Example B 1

Aquifer at -3 m (1.20 m below drain level)

h = 0.800 h / q = 400

q = 0.002 8h/q = 3200

K = 0.50 D2 = 1.20 K2D2=0.60 Lo = 402 m L = L - c = 3 1 3 m

K = I O D3 = I O K3d3 = 50 c = 8 9 m SPC: L = 305 m 2

3

K /K = 20 D /D =4 KD = 50.6 c < 0.3 Lo 3 2 3 2

a = 4.0 (c/L0=O.22)

Note: The fZow aboue the drains can be neglected i n t h i s drainage s i tua t ion KD aD2 c = - In __ K >> K2). Therefore, KD = K D + K3D3, whereas

3 2 2 K2

3 If KD-values are high (here KD = 50 m /day), the flow in the K2D2 layer can also

be neglected and KD = K3D3 , the more so because the KD-value of the aquifer is a very approximate value.

The L-value can be determined in two ways: either by using Graph I or by using

Eq.(9c) in combination with an SPC. It is recommended that both methods be used

to allow a check on any calculation errors. Small differences may occur in the

results of the two methods, but this is of no practical importance.

Example B l a

Drainage by pipe drains (u=0.30 m), instead of ditches

Lo = 402 m (see Ex.B I )

c = 277 m

c/Lo = 0.69

LILo = 0 . 4 5 + L = 180 m

Note t h a t i n t h i s s i t ua t ion the use of pipe drains instead of ditches has a

great inf luence on the re su l t i ng drain spacing.

Example B ‘2

Ditch bottom in the aquifer (u=2 m); drain level -3 m.

h = 2.000 8 h/q = 8.000 ~ ~ = 6 3 2 m L = L - c = 6 2 7 m

q = 0.002 KD = 50 c = 5 m

33

Example B Za

Pipe drain in the aquifer (u=0.30 m), drain level -3 m.

Lo = 6 3 2 m (see Ex.B 2)

c = 1 4 m

L = L - c = 6 1 8 m

Note t h a t i n t h i s s i t ua t ion the use of pipe drains instead of ditches has very Z i t t l e influence on the drain spacing, because here the radial resis tance i s very small.

Example B 3

The aquifer at - 8 m (6.20 m below drain depth) ; KD = 50 m2/day;

ditch (u = 2 m); drain level -1.80 m.

h = 0.800 h/q = 400

q = 0.002 8q/h =3200

K2 = 0.50 D2=6.20 K2D2 = 3.10 Lo = 412 m c/L =0.61

K3 = 10.0 D3 = 5.0 K3D3 = 50.0 c = 253 m L/Lo=0.49 + L=202 m

K K =20 D3/D2=0.8 KD = 53.1 3 2

a = 3.5

C .

Example C 1

In f l uence of t h e KD-value o f an a q u i f e r

In Example B 1 (aquifer at -3 m, KD-value = 50 m2/day + L = 305 m, the

KD-value has been estimated from borings to be at least 50 m2/day. Now the

question arises whether it is worthwhile to carry out pumping tests to obtain

a better estimate.

If, in a certain drainage situation, one wants to analyse the influence of

the magnitude of the KD-value on the spacing, it is convenient to calculate

firstly, - In * I KP

, which in this case equals 1.75.

34

Assume KD = 100

8 h/q = 3,200 Lo = 566 c / L o = 0.31

KD = 100 c = 100 x 1.75 = 175 L = L - c = 300 m

KD = 500 + L = 570 m

KD = 1000 + L = 625 m

These computations show that in this case it will indeed be worthwhile to

carry out pumping tests.

Example C 2

In Example B 3 , with the depth of the aquifer at -8 m and KD = 5 0 , L = 200 m.

Making the same computations as for Example C 1, we get:

8 h/q = 3200

KD = 100 c = 504 m L / L ~ = 0 . 3 8 L = 215 m

Lo = 566 m c/Lo = 0 . 8 9

~~

KD = 1000

8 h/q = 3200 Lo = 1789 m c/Lo = 2.82 SPC and Eq.9

TTLt L = - = 250 m L = 245 m

8c KD = 1000 c = 5040 m

These results show that here an estimate of the KD-values will suffice and

therefore - in contrast to Example B 1 - no pumping tests are required.

Importance of geohydroZogicaZ investigations

If we compare Situation B 3 with that of A 2 (Fig.4), we get:

A 2 : impervious layer at -8 m + L = 100 m

B 2 : instead of an impervious layer, an aquifer at -8 m + L 200 m

35

This comparison of drain spacings shows clearly that a drain spacing can be

considerably influenced by layers beyond the reach of a soil auger.

From Fig.4 it will be clear that if in the given situation drainage investi-

gations are only conducted to a depth of 2 m and a barrier at 3 m is assumed, the

recommending drain spacing will be 50 m.

If, however, geo-hydrological investigations are conducted, they will reveal that parts of the area can be drained with spacings of 300 m (drain level -1.8 m)

or 600 m if the drain level is -3 m.

36

4.2 Summary of graphs and equations

G r a p h s

G . 1

G . I a : K / K - and D / D -values + a-value

: c/Lo-, B- and L / L -values + L-values ( d r a i n spac ing)

( a u x i l i a r y graph f o r

r a d i a l r e s i s t a n c e ) 3 2 3 2

G . 1 1 : Homogeneous s o i l and p ipe d r a i n s -f L-value ( f o r a l l D - and 2 K -va lues) 2

( a u x i l i a r y graph i f a SPC i s not a v a i l a b l e ) D L G.III: D I n 2 o r L I n - 2 u

E q u a t i o n s

Only one perv ious l a y e r below d r a i n depth

U S E D < $ L 2

Eq.(3a) L2 + E L I n r_2 - 8 KD h /q = O o r i g i n a l e q u a t i o n out of u s e TlK- U L

E q . ( 7 ) (t7 + (so) (kr- - B ($)=O g e n e r a l i z e d eq. G . 1 O O

2 8 D T l 2 u

E q . ( 8 ) L + - L D I n - 8 KD h / q = O modif ied e q u a t i o n SPC

Eq.(lO) L = L - c s i m p l i f i e d eq. f o r c <0 .3 B 10.1

where Lo = 8 KD h /q D

2 u c = D I n 1

L Eq.(ll) L I n - = n K 2 h/q G . 1 1 1 o r

SPC ~~

Two perv ious l a y e r s below d r a i n depth

2 8 KD Eq.(9) L + - L - I n a - 8 KD h / q = O

T K 2 u G . 1 o r SPC

where KD = K D + K2D2 + K3D3 o r f o r K >> K 2 : G.1a 3

KD = K D

a = f ( K ~ / K * , D ~ / D ~ )

+ K3D3 2 2

37

4.3 Programmes Scientific Pocket Calculator (SPC)

Note: These p r o g r m e s should be adjusted i f necessary, t o suite the spec i f i c

D 2 8 KD a D 2 Eq. ( 9 ) : L +- L - In - - 8KD h/q = O E q . (8): L 2 s LD2 In 2 - 8KD h/q=O

KD = K D + K2D2 1 1

4 b Z - D D I n 2

71 U v Kz

KD = K2D2 + K D 3 3

u KD a D 2 i b - _ I n __ TI T K7 U

Programme examples

KD ENT 8 h /q (x) KD ENT STO 8 h /q (X)

D2 ENT u (+) ( I n ) o r T ( + ) u ( + ) ( l n )

D2 ( X I 4 ( X I n (+)

STO ENT ( X ) (+) (&) RCL (-) STO ENT (x) (+) (&) RCL (-)

a ENT D2 (X ) u (+) ( I n )

RCL (X) K2 (+) 4 (X ) TI (+)

or

D2 = 5 KD = 4.24 a = 4.6 KD = 17 .3

u = a4 8 h/q = 2.400 L = 8 7 . 0 7

D2 = 1.6 K = 1 . 2 L = 7 3 . 2 2 2

u = nr 8 h/q = 800

r = 0.10

38

Appendix A. Derivation of the generalized equation of Ernst

The basic equation (Eq.6, Section 3.2) reads

8KDh 8KiDih 8 K2D2h

Multiplying all terms by ___ L2 8 KzD2h

gives

Multiplying by I I I ‘ L I 1 -

L - I l I Lo i Lo

L and setting - = x, we get Lo

KiDi + KPDP KD aD 2

and - _ _ . , D2 In - = c KiDi

writing - K2D2 + = K2D2 KzDz U

K2D2 multiplying all terms with - KD yields the final equation

x3 + [k] x2 - x - B [ T] 8c = O

where

39

I f we compare t h e above b a s i c e q u a t i o n w i t h t h e Hooghoudt e q u a t i o n and we assume

t h a t b o t h e q u a t i o n s y i e l d t h e same r e s u l t , t h e n

Dz 8 D 2 D2

1 +-ln- V L U

d =

where u = Tr

From a comparison o f t h e d-value o b t a i n e d by u s i n g t h i s e q u a t i o n and

Hooghoudt 's d - t a b l e f o r r = 0.10 m , i t appea red t h a t i n m o s t c a s e s t h e g r e a t e s t

s i m i l a r i t y was o b t a i n e d by u s i n g

Dz -irr i n s t e a d o f - , where a l s o t h e u s e of u = 4 r gave b e t t e r r e s u l t s t h a n u = n r .

I t s h o u l d b e n o t e d t h a t t h i s i s o n l y o f t h e o r e t i c a l impor t ance . F o r r e a s o n s

of conven ience t h e a u t h o r p r e f e r s t h e u s e o f u = 4 r .

40

Appendix B. Layered soil below drains

a/h =0.8

D /D =0.25 3 2 K /K

F i g . 5 . Comparison of caZcuZated &din spacings based on the equation and graph of Ernst and on 36 graphs prepared by Toksöz and Kirkham (1971)

0 . 4 0 . 2 O

1 .5 4 u?

NOTATION ERNST NOTATiON KIRKHAM

l l i l i i l i i i l l i l i l R l l l i l

I K3 I

0.02; .O2 I

I I

I

.I01 . I 2

.201 .24

.501 - 6 0 I

K >

0.1 K2

D + 0 . 4 0 2 .4 6 . 4 m 3

36 .0 56.0 3 6 . 4 36.C 36 .8 3 6 . 8 36.9 3 C . R

3 6 . 4 36 .8 3 7 . 9 3 R . U 39.9 J9.0 4 1 . 0 4 2 . 0

3 6 . 8 36.8 3 9 . 7 40.0 4 3 . 4 42 .0 4 5 . 7 46.0

3 7 . 7 36.8 4 4 . 4 45.0 5 1 . 3 50.0 55.7 56.; ~~ _ - ~-

4 2 . 4 43.0 5 9 . 7 59.0

4 9 . 4 48.0 7 5 . 7 74.1)

57.7 56.0 8 9 . 8 90.0

8 4 . 0 D Z . 0 112.5 112.0

125.0 123.2 125.0 123.2

5 0 160

7 3 . 2 72 .0

9 1 . 3 90.0

103.1 102.0

1 1 9 . 3 118.0

125.0 123.2

8 5 . 6 8 3 . 1

103.9 7 0 7 . J

112.7 112.0

121.6 122.0

125.0 123.0

41

Procedure

For the type of calculations given above (many values: some variable, some

fixed) the following procedure is recommended:

1 ) determine the fixed values, which are here:

aD 2 D2 U m K2D2 = 1.92; ln - = In a + In - = In a + 1.63

2 ) calculate the various KD-values (KD = K3D3 + 1.92) and determine the a-values (Eq.la); write down these values and use the required con-

sistency in the rows of figures as a control for their correctness.

3 ) make a program for the available SPC, based on Eq.9 and the constant

values

42

Appendix C 1. Construction of Graph 11, based on Hooghoudt's table for r = 0.10 m

0.125L2x10-2 L 2 = or for KI = K2, L = 8Kd' h /q K = L2 K = -

8 K ~ d h t 8 K l d l h

q

L= O . 1 2 5 L 2 ~ 1 0 - 2 = .̂

I . d=

d ' =

K=

2 . d=

d ' =

K=

3 . d= d ' =

K=

5 . d= d ' =

I K=

IO. d=

d ' =

K=

c w

d i = d t Di= d t 0.5h h /q = 100

10 15 20 30 40 50 75 100 150

.125 .281 .50 1.125 2.0 3.125 7.03 12.5 28.125 50.0

200 ( m )

.49 .49 .49 .50 .50 .50 .50

.79 .79 .79 .EO .80 .80 .80

. I 5 8 .356 .633 1.141 2.50 3.91 8.79

.80 .86 .89 .93 .96 .96 .97 .98

1.10 1.16 1.19 1.23 1.26 1 .26 1.27 1.28

. I14 .242 .420 .915 1.59 2 .48 5.58 9.76

1 .O8 1.28 I .41 1.57 1.66 1.72 I .80 1.85

I .38 I .58 I .71 1 .87 1.96 2.02 2.10 2.15

.o91 . I 7 8 .292 .602 1.02 1.55 3.35 5.81

1.13 I .45 I .67 I .97 2.16 2.29 2.49 2.60 2.71

I .43 1.75 1.97 2.27 2.46 2.59 2 .79 2 .50 3.02

.O87 . I 6 0 .254 .496 .813 1.21 2 .52 4.31 9 . 3 1

I .88 2 . 3 8 2.75 3 .02 3.49 3 . 7 8 4 .12

2 . 1 8 2.68 3.05 3 .32 3 .79 4 .08 4 .42

.229 .420 .656 .941 1.86 3.06 6.36

2.57 3 .23 3 .74 4.74 5.47 6 . 4 5 7 . 0 9

2.87 3 .53 4 .04 5 . 0 4 5.77 6 . 7 5 7 . 3 9

.392 .567 .774 I .39 2.17 4.17 6.76

2 .58 3 .24 3 .88 5 .38 6 . 8 2 9.55 12.20

2 .88 3.54 4 . 1 8 5 . 6 8 7 .12 9.85 12.50

.391 .565 .748 1.24 1.76 2.86 4 . 0

o - m m - o m . . . . I D N

F.

o10 o . i 1 0 - . .

m

O

O

I,

O

II

a -

/I o Y m

-co L o m O 0

+ + . .

N N n a I1 I1

a

E O d

E O o I,

. N O *

I, I/

L 3

3

O O o m

0 3 /I

/I m , 5 r

1 c m

3 V \ E E

o w O 0 w o

O 0

I1 I,

c 5

. .

co i- u m 'il

m m o m v, r . . . . .

L n N m oom . . . . m - LP

0 0

N O U

. . .i

U u)

. r - .?a

T i m i - N m o m o m NJ m . . .

. N N . 3 Ln

m m

O m m . .

m rl

O m h m m o m . . . . - N m

a m m o - - m o m - . . . C e o \

r. . . -

m .* ID u ) m o m . . . .

N N ' O - 0

m N 10 -f

- - o m . . . . u) -

a o c N 1 0 C

N . . . -

___._

.* m - c o o . . .

m

. - N

m m N - N o m . . . .

O -

cj U N

m

m m o N

~

O m

- O m O

O m

N

O N

44

“)

o m N r. m m o

m ? - m -4

- m - 0 N

. . m

h Ln

- m O -

“1

“7 h N Ln “ O m I\

u!

. . . . h m - m

ul h

’i,

o m u 3 m m

m m n a m

m - 0 1

m

. . . . 0 0

m

h o - a ~n o e - m w . . . . O m N

O O N

O m - O O - m r.

O m

O v

I1

el

U O

, i:

N

2 I1

r l l s C

rl

i

8

N

m v

m - m

-, o ~n o m . .-“u! “1 . . . . PI) m v L-

N u3

- i D m N

O m

m

m m , - o

N N

in O

iD

m iD

N - m - N m

\D O m O 01 O

O o

0

O o

L r l

O m

O - O m

O

m O

O -

4 5

List of symbols Symbol Description Dimension

a

B

C

d

D l

D2

D3

DV

h

K2

K3

KV

KD

L

q

91

42 r

U

46

geometry f a c t o r f o r r a d i a l f low depending on t h e h y d r a u l i c s i t u a t i o n d imens ionless

t h e f low above t h e d r a i n a s a f r a c t i o n of t h e t o t a l h o r i z o n t a l f low = K D f K D dimensionless

r a d i a l r e s i s t a n c e f a c t o r m (meters)

t h i c k n e s s of t h e e q u i v a l e n t l a y e r of Hooghoudt

average depth of f low r e g i o n above d r a i n l e v e l

t h i c k n e s s of t h e nerv ious s o i l l a v e r below d r a i n

1 1

l e v e l = c r o s s - s e c t i o n a l a r e a of f low a t r i g h t angles t o t h e d i r e c t i o n of f low p e r u n i t l e n g t h (m) of d r a i n (m2 /m)

m

t h i c k n e s s of t h e perv ious l a y e r , i f any, below l a y e r D

t h i c k n e s s of l a y e r over which v e r t i c a l f low i s cons idered

2

h y d r a u l i c head = t h e h e i g h t of t h e water t a b l e above d r a i n l e v e l midway between t h e d r a i n s

h y d r a u l i c c o n d u c t i v i t y (h .c . ) of t h e s o i l ( f low reg ion) above d r a i n l e v e l

h . c . below d r a i n l e v e l ( l a y e r D )

h . c . of l a y e r D

h .c . f o r v e r t i c a l flow

2

3

t h e sum of t h e product of t h e p e r m e a b i l i t y (K) and t h i c k n e s s (D) of t h e v a r i o u s l a y e r s f o r t h e h o r i - z o n t a l f low component accord ing t o t h e h y d r a u l i c s i t u a t i o n

d r a i n spac ing

d r a i n d i s c h a r g e r a t e per u n i t s u r f a c e a r e a p e r u n i t time

d i s c h a r g e r a t e of t h e f low above d r a i n l e v e l

d i s c h a r g e ra te of t h e f low below d r a i n l e v e l

r a d i u s of t h e d r a i n

wet ted per imeter of t h e d r a i n

m

m

m

m2/day

m

(m3 p e r day/m2) m/day

mf day

d d a y

m

m

References

DUMM, L.D. 1960. Validity and use of the transient-flow concept in sub-surface

drainage. Paper presented before ASAE Meeting, Memphis, Tennessee,

Dec. 4-7.

ERNST, L.F. 1962. Grondwaterstromingen in de verzadigde zone en hun berekening

bij aanwezigheid van horizontale evenwijdige open leidingen. Versl.Landb.0nderz. No.67. 15 (English summary)

ERNST L . F . 1976. Second and third degree equations for the determination of

appropriate spacings between parallel drainage channels. J.of Hydrology

(in preparation).

HOOGHOUDT, S.B. 1940. Bijdragen tot de kennis van enige natuurkundige groot-

heden van de grond. Versl.Landb.Onderz.No.46. (14) B.

MAASLAND, M. 1956. The relationship between permeability and the discharge,depth

and spacing of the drains. Bul1.No.l.Groundwater and drainage series. Water

Cons.and 1rr.Comm.New South Wales, Austr.

TOKSOZ S. and DON KIRKHAM. 1971. Steady drainage in layered soils. 11. Nomo-

graphs. .T.Irr.& Drainage Div.ASCE. Vo1.97, pp . 19-37.

VAN BEERS, W.F. 1965. Some nomographs for the calculation of drainage

spacings. BulL.No.8. ILRI, Wageningen.

WESSELING, J. 1973. Subsurface flow into drains. Publ.No.16. Vol.11.

ILRI, Wageningen.

47

L I S T O F A V A I L A B L E P U B L I C A T I O N S

P U B LI C A T I O N S (3/F) ti-tch H . Jacobi, Riv"mhrc~nii~nt cn Europe. 1959, 152 pp. (3/D) Erich H. Jacobi. Flirrhiwinigung in Europcr. 1961. 157 pp. (6) A priorilj, .\chi~me for Dut ih lund consíiliilulion projci~ts. 1960. 84 pp.

cmrnl ofmvi'sîment.s in lundr i41mat ion from thep(iint o/ v i i ~ ~ . of / h i ~ n u f i o n u l ~ ' i ononij'. 1969. 65 pp. ga. Lncul udmini.strution o/ M ? U I P ~ i,ontrol in u numher (if Europeun counlries 1960. 46 pp.

(9) L. t:. Kamps,Mud dili\trihurion und lund redirmution in !he iwstern Wudden Shi i lk i~*s . 1963. 9 I pp. ( 1 I ) P. J . Dielenian. etc. Redumation o/ s u l l u/frNri ted .coil.\ in Iraq. 1963. I75 pp. (12) f. H . Etlel~nan. AppIicciti~~n.\ nf soil.survry in lundclfvrlopmrnt in Europc'. 1963. 43 pp. ( 1 3 ) L. I . Pons, and I . S. Zonnevcld. Soil riprning urirl.roil i~lris.si/ïcutiiiri. 1965. 128 pp. ( 14) G . A. W. v;in Je Goor. and G. Zijlstra. lrrigution requircmen/s fo r riouhlc cropping of lowlundrice in Mulqvo.

1968. 68 pp. (15) [l. B. W. M . v a n Dusseldorp. Plunningo/.srrvii~c í 'r i i trcs in ri~rulureusof~J~,vc,lo/iing countrie.s. 1971. I59 pp. (16) /)ruinuge principlrs und irpplicii~inn,~. Vols l / l V (1972~ 1974). 1455 pp. ( 1 7) Land evuluotion fo r rurcil p~irpo,sc,s. 1973. I 16 pp. (19) M . G 1305. and J . Nugteren O n irrrgrrlion e//ici(wiie\. 1974 89 pp. (20) M . G. Bos. ed. lIi,\ilior~i, h.Ierrwr.criwii . S ' i r . t r i ~ i i w i ~ \ . I Y76. 464 pp.

B U L L E T I N S (I) ( l /D) W. F. J. van Beers. Die Bohrloch-Methode. 1962. 32 pp. (3) (4) (5)

( 6 ) (7) (8/F) W. F. J. van Beers. Quelynr'.~ non7ogrcimmr~.s pour le cd i , i r l (/i,.\ espuci'mrwt.r di2.r rlruins. 1966. 2 I pp. (9) (IO) (I I ) (I I/F) G. P Kruseman, and N A . de Ridder. Int' ( I I / S ) G. P Kruseman, and N. A. de Ridder. An (12) J. G. van Alphen, and F. de los Rios Ro ( I 3) J . H . Edelman. Groundwutc,r hydruulii,.s of'c.vîiv?.\iw 11q~ifors . 1972. 216 pp. (14) Ch. A. P. Takes. Lund.wtt/i~mmt rinilrr.\cItlf~mí~rirproiri~t.s. 1975. 44 pp. (15) W. F. J . van Beers. C'omputing drnni cpui.inlga. 1976. 47 pp.

W. F. J. van Beers. Thr uirgiv hole mcthod. 1958. 32 pp. Rev. offprint 1970.

W. F. J . van Beers. Acid Sulphote Soil,\. 1962. 31 pp. R. Verhoeven. On î h e culcinnii~urhonuti, c'ontewî of ~ o u n p murinc .sediment.\. I Y63 27 pp. P. J . Dieleman and N. A. de Ridder. Studi/ic.r of.sul1 wuti'r movcmenl in /he Bol Guini Poliler, ('Iiud RqJuhli(,. 1Y64. 40 pp. A . J . de Groot. Mud irim.spnrl stucik% 117 i~ou.stu/ ivuîi'rs /rim I h c wrslern S~~hcl í / l lo Ihc Dawi.sl7 Fruntiiv. 1964 Cbde of practice f o r the i ks iRn n f ' o p i ~ ~ wuteri"ir.s~~.s enid ancillurj> .struc'turc.\. 1964. 80 pp.

D J. Shaw. The milnugil South- Wcstiw cxtmsiiin î o /hi, Gciiru .Si hemr. I Y65 37 pp. t; H o m m a . A viscous fluid moilel fhr dwmn,strufion o/ groundwutrr /h to purullc4 (lrui17,\. 1968. 32 pp. G. P. Kruseman. and N. A. de Ridder. Aiiu1.ysr.s undc~viiluririon ofpimiping / P . s ~ rliitu. 1970. 2nd ed. 200 pp.

n C I iliscu.s.sion ilesp~~mpugc.~ d ' imui . 1973. 2nd ed 21 3 pp. uluucirin de ki.5 i k i ~ o a di, eii.siryo.\ por hornhro. 1975. 2 I2 pp. 'p.cilerous s ( n I s . 197 I . 44 pp.

(16) C . A Alva. J . <i. \'ai1 Alpheil et a l l'rohli'nl(/.\ í h i l l ~ í ~ i ~ o / i ~ , ~ ~ \u l i~ i i í l c id i~ i~ I ( / ( ' ( I \ I o 1'cI.uioIo 1'976. I l h p p

B I B L I O G R A P H I E S (4)

( 5 ) (6)

L. F. Ahell, and W . J . Gelderman. Aniuilutrd hih l iogrupl~~~~ on rei lum[ition c n i d inipriivmiiwl of . s u / i n i , und ulkuli .\oil.s. 1964. 59 pp. C . A. de Vries, and R. C . P. H. van Baak. Drufnugc n/ ugrii~ulrurcil / ( n ? d s . 1966. 28 pp. J. G. vali Alphen, and L. t; Ahcll. Ainrotuti~d h i h l i i i ~ r o p h ~ ~ on rci~li~mirt~on uni1 inipro~~í'nicnt of . s c d i n c uni1 .sodic .soil,\. (1966 1960). 43 pp.

. A. dc Vrics. ,4prii~iItiirul c.rtcriaion in diwdíipinr: (outirr.ic\. I25 pp. J. Brouwer, and L F. Ahell. Bihliographv o11 u i l î o n irrilgu~inn. 1970. 41 pp. Raadsma, and G. Schrale. A n n c ~ f ~ t c i l h i h l i o g r u ~ ~ h j ~ on .\ur/at.s irrigu/ioii mi,îhoi/.\ 1971. 72 pp.

(10) R . H . Brook. S ~ i l \ur nti'rprc!uîiiin. 197.5 64 pp. (I 1) (12)

A N N U A L R E P O R T S free of charge

Lund und wu~i'r drvclopmi~nt. 1975. X0 pp. Lunrl uni1 wuIcr i/i~vi~lnpnic~wr. 1976. 96 pp.

Infot-ination ahout exchange and w l c 01' ILK1 puhlic;iiion\ can lx obtained from

INTERNATIONAL INSTITIJTE FOR L A N D RECLAMATION A N D IMPROVEMENT/ILRI P.O. BOX 45 W A G E N I N G E N i T H E NETHERLANDS

GRAPH I Determination of drainspacing with t h e generalized Ernst equation

GRAPH I Determination of drainspacing with the generalized Ernst equation 1.0

1

0.9

O. 8

0.7

0.6

o .I 0.2

O .3

Graph I a : Equivalent layer (aD2) for radial resistance ;

soil below t h e drains consists of two pervious layers

0.1 0.15 0.2 0.3 0.4 0.5 0.6 0.8 1 1.5 2 3 4 5 6 7 8 9 1 0 15 2 0 30 40 50

K3’K2 -

GRAPH II Homogeneous soil and pipe drains (rz0.10m) L in m

D L Graph : Auxiliary graph for D In or L In

D L Graph III : Auxiliary graph for D In o r L In

+ L

6000

5000

4000

3000

2000

1000

800

600

400

400

3 0 0

200

D In - U (C)

10

6

5

4

3

2

1

1 2 3 4 5 6 t 8 10

D D I n , U

600 500

400

300

200

1 O 0

80

60

5 0

40

3 0

20

10

2 0 1 30 40 50 60 00100

L